Data-driven spectral analysis of the Koopman operator

By Milan Korda, Mihai Putinar, Igor Mezi'c. Applied and Computational Harmonic Analysis, 2017

138 citations recorded by the source corpus.

Starting from measured data, we develop a method to compute the fine structure of the spectrum of the Koopman operator with rigorous convergence guarantees. The method is based on the observation that, in the measure-preserving ergodic setting, the moments of the spectral measure associated to a given observable are computable from a single trajectory of this observable. Having finitely many moments available, we use the classical Christoffel-Darboux kernel to separate the atomic and absolutely continuous parts of the spectrum, supported by convergence guarantees as the number of moments tends to infinity. In addition, we propose a technique to detect the singular continuous part of the spectrum as well as two methods to approximate the spectral measure with guaranteed convergence in the weak topology, irrespective of whether the singular continuous part is present or not. The proposed method is simple to implement and readily applicable to large-scale systems since the computational complexity is dominated by inverting an $N\times N$ Hermitian positive-definite Toeplitz matrix, where $N$ is the number of moments, for which efficient and numerically stable algorithms exist; in particular, the complexity of the approach is independent of the dimension of the underlying state-space. We also show how to compute, from measured data, the spectral projection on a given segment of the unit circle, allowing us to obtain a finite-dimensional approximation of the operator that explicitly takes into account the point and continuous parts of the spectrum. Finally, we describe a relationship between the proposed method and the so-called Hankel Dynamic Mode Decomposition, providing new insights into the behavior of the eigenvalues of the Hankel DMD operator. A number of numerical examples illustrate the approach, including a study of the spectrum of the lid-driven two-dimensional cavity flow.

Subfields: Electrical and Electronic Engineering, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4fb386490bc0f796d24a61801d58952b8957dcdc

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Electrical and Electronic Engineering

Igor Mezi'c

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders (2020, background): “The identification of a reduced basis to ensure a compressed representation that is minimally lossy is a core component of most ROM development strategies (some examples include [25, 26, 27]).”
On Numerical Approximations of the Koopman Operator (2020, background): “Continuous spectrum is effectively dealt with in [10,49] using harmonic analysis and periodic approximation methods, respectively.”